thq_05 - O A05/015/01 0 Use T.C 16s MATH 2451 thq_05 Mult...

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Unformatted text preview: O. A05/015/01 0 Use T.C. 16s MATH 2451' thq_05 Mult Exam ble Calculus 1var1a Course Due Date 2016-o2~o4 (Thu): McCary Instructor . Net ID e W m 0 AI @ @ ® @ ewe [email protected] ® @ «my O @ me E, as . _ . ® ® 3 . [email protected]@@©©@@®@@@@©eaeeweeeee®ee®eeeeaa [email protected]@@@®@®@C®®®seemsas Ix 1.. _ 3 «ma ® Ame [email protected] weeeeeeeeweeee my 0 ® @ E ® @ Amy © me ® @ © Ame /i flee efieeeeeeeee®ea m ‘\l ‘ .I II my E O ® ® ® ® .. , .l 3 ® ® ® ® ® ® @ @ ma ® © @ ® @ ® L @ ® ® @ @ aw @ mt, «UV m w @ fl. @ ® ® (First Name) xU/ a“ .. raw A,..\ Q7/1986 ®@@@@@®®@ saeeeea k )1 me, Q @H (Last Name) 07 - 2018-01-28 14 .. A05/015/02 . 19112::vn2- 1. (10 points) Find all values of a. and b which make the function differentiable zit all points. You must use left ane limits= the limit definitibn of continuity at a. point, and the limit definition of derivative at a point. PGF‘WN I110"! meeég A’T—D— . éamfi'é 5r: Ifirl up; 1*0121‘22 fiT§§;?Jz§JT";4véé— f() am2+b$+4 I .LI‘JT-C 'i' m = ‘-—’ - a$+b 2:22,! a, fi<”>j;_ FM? vflpmé‘; as AWL" Fm- ifiég} é 90M;.—-_iv.g,d_; | W39 a"); fim-Jrlbflt (Lew? 2* 4a L; M 73):; 2am Mtge}: "' 'r' ' V» a 5.2. N 9157‘) léL/JNHWM'C’l/LS W V/Y' plF‘F‘éfl/‘efj? ’f’kréi'f WWW emf—iaihfirrj/xr b - ' ' 4.6-1? : 2A -% L; —7 #4} H _' f5“ 1 I__ h / )0 '1 Lari M:- EA 1'. 7—? "’ 1/; I-fii—‘g i‘;':,.:-“;. DN/ Irv aflr‘ffi F3»? '-FL":‘~'1'_,} T5 ‘33 _'2"’«: '41:- . 5.1-». in z: -:_-.~r:v°c;a,-:/ Km (49' . fly “CW? WWW-9 feméAwp ll. LAM-HE; Mmfi MATCH- A1— 15; i :- _ a» \ . ' '[email protected]—/ ‘4 '1 H— 11. .. 6012’; putt-*gz-ig’N’IM-BLE’ Afar; 1 l9 1 '- l2";— FF: iGraderOnlyJr O 069 o O. A05/015/OL O _;-0 points) n . ,— n , . (a) Compute frag-gm) using the limit definition of partial derivative. 1 lie-i L7,: (b) Compute using the limit definition of partial derivative, "7 In 3 \ r F r ‘ (c) Show f is not continuous at (l. ,. ‘f- l- ,‘“ y 0 otherwisezfiuéql “A? w W. ‘ ’Y 1;": ih};_..., # _ E. l is)? gaff ages {are 117 + 11'») 5 (>554 7: reef—- fig, f K lflv‘fi 0 "' (a)?! ,x—y \. v 965 "N w in“, C fiDMIrvl/LH ‘lmy'aLté‘Si 1i“ K t ht game h’l/l fill i 42(0):?) lt wrap If Dir-7“] f i I ‘ I “265) r -%Q" :15 HPLQIQET’U :‘0 i If l" PJ lg NOT ”' (GN'ilL‘VL'1‘45’i‘: "g1 g Q “ \ ' n - a ‘ ‘ V“ f ' :7“ . 7 1, __ / V - i“ _ .* ti! A ._ .4 ‘ - a *U " “477171.41?- ' ‘- ‘ J J .1 %, \3'"Lll-'lr'\hlzc'w L M"- -. J/lW’l Qt: T: 0 M” “’6‘ ’ ' A " 1 ‘ 5' w, " ~——/\_._——-——~ The purgste of this exercise is to illustrate one of the weaknesses of partial derivatives: even if all partials exist at a, point, the function may still be discontinuous! This is in contrast to scalar calculus Where the existence of g’(a) implies continuity at a. See the Comment about symbolic calculators at the end of the next exercise, which also holds for this exercise. (fl - ‘k .1 J, Grader Only J, 6 Pea-2n £4, 9‘ ‘ o o A o 5 / 0 1 5 / 0,4 9 {£99549 ores-a2 W” WA ‘H 17 C?” rape”, F _. ‘_ ~ ‘ ‘ 'nfif cgefinitio‘ff oi‘partial derivative. / I j _ '_.- 1“ 'u'n -‘ Ir'm'il I .- ‘ - i" "/43 7H ' ' I. I my :1: y y / V' .‘ m #5 inf/.7" / ./ f 2 3 +9 / 0 otherwise 3. (10 points) Find 3—56), $503), $1346), and gays) usingith— ’ r c, -. r .. " ,3 ’f —— 7—\ \7 '3- . «If " W's) (WW/F" ’7' 92—» “’1 . :" 74 7 add“ ‘1‘ j ——-'>< Pa”- ..I-ix’“ in iv ; ‘7' £1330 Hurts‘LéZQhW-f} ' "fizzfiwf fax? 7 _ d Z I...‘ ‘ J “I: _ I E :3 is}? I \ z: ' [4375’] «HA? 3+9. 7—75? 3]“: a an 4' films? 2'; W V; H“ Jaf 4 4 2:. ,5 y, :12} 4 f g} Liege - m1— e—Y‘aii‘n—“v ‘2’ lf‘erm a . roe/+21% page — 7‘anch 143771- 2137 Baldw- ”+-an+121,1~,? "l—L 4- ; 2 7— 4 r fl -} 3 3:7 7 a r” A 5 *2"? 7’“ “H4 ‘47—”‘7‘jl W “1/7" itiygfli’sflflwifi)’ — g 7 3 3 it}??? 411:“; 3 7 +1.? 3‘" xii/75%“ r,” 1m: 7 a» L (“A ,{7 asleewesna_, W”) L" "Q “irreftyz’l We figfitésfiluéh%a w 2 . W 5 flush-figure)‘er—W gran-t - ‘L‘m’tr‘f’ri‘i’ at (WW/i); (“flfiflflmeawr 14 mph/7):, #0 i {94" 9‘15? . . . . . . ” ./ I ‘ . . The purpose of th1s exerc1se is to show that mixed partlals 1n dgi'grentporders are not always equal- Don‘t get lost 1n the limit formalism — try pier-ting assisfifia second partials onfilfi‘am Alpha (or, even better, download SAGE Math and start using it, MATH majors). Can you see why the mixed partials are not equal? Feel free to use a symbolic calculator to compute the partials of f away from 5, but remember that you must use the limit definition to compute the partials at 6 because 5 is not in the domain of that rational expression. l Grader Only J, . O® @©@@@@©®@@® . O. A05/OlS/05 0 ,0 points) Compute the Jacobian. You must Show your work and not use a symbolic calculator. I. (all in -' J 33;? r énvoéf (3 "._ l 7." i 2-way “-‘ ’ ' z / . lQX‘r/s '; 1 '60?" * I l /r ’ ‘__ f [ J, Grader Only J, .. A05/015/06 . 5. (10 points) Let f = Show, using limits, that for any number m, 5%(i(o+h)—r(0)—mh)=0 3;}: 3x23 (” VA?“ 'r-J Q r>< but f1}i_rrb%(f(0+h)—f(0)—mh)=0 is never true: there is no m such that mh is a good approximation to the increment A f at U. (rgfo+-t)-£(5)m)j )om mt mime/mt e i. m. t w— My \‘F \A «-----><', Eli-1X u we in ‘. r W-Ji‘h Mil/i) : p/M’W "IA" m m z- Wm) :- O [/1':?OH‘ lflfir 0:! , lflaofi ‘ LlMl'r' Ewe—rg__,«i{r><- la; CflN—WNMGMS Aron _ M 7w» \/ WWW» Mt :0 [/80 f {£{Ui Life 1? ‘3‘}- Iii”; i: -},V,lril/El Z“: ,'_‘/V‘1 W1 \fiflPQH/l/‘r Maj—min )Jim fog/1N hm) = -I _ . "m I” rm UM” VINE hog/f .- f, w 4m H'Q\/(/THé AEEO Luff}? V/XLMC TbtpV"G| loN l5 00mm 0m” AT 0, E: :T’ 1:; N55” mesa/126m mate The purpose of this exercise is to wean you off of the scalar calculus definition of derivative and to get you accustomed to the Fréohet definition- iGraderOnlyJ, . O® @@@®@©©®®®Q . ...
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